Quantitative Analysis of Diffusion Equation When Diffusivity is Given as Function of Concentration

Discrete filtering of information over triangulated surfaces has proved very useful in computer graphics applications. This technique is based on diffusion equations and has been extensively applied to image processing, harmonic map regularization and texture generating, etc. [C. L. Bajaj and G. Xu, ACM Trans. Graph., 22 (2003), pp. 4-32], [C. Wu, J. Deng, and F. Chen, IEEE Trans. Vis. Comput. Graph., 14 (2008), pp. 666-679]. However, little has been done on analysis (especially quantitative analysis) of the behavior of these filtering procedures. Since in applications mesh surfaces can be of arbitrary topology and the filtering can be nonlinear and even anisotropic, the analysis of the quantitative behavior is a very difficult issue. In this paper, we first present the discrete linear, nonlinear, and anisotropic filtering schemes via discretizing diffusion equations with appropriately defined differential operators on triangulated surfaces, and then use concepts of discrete scale-spaces to describe these filtering procedures and analyze their properties respectively. Scale-space properties such as existence and uniqueness, continuous dependence on initial value, discrete semigroup property, grey level shift invariance and conservation of total grey level, information reduction (also known as topology simplification), and constant limit behavior have been proved. In particular, the information reduction property is analyzed by eigenvalue and eigenvector analysis of matrices. Different from the direct observation of the local filtering to the diffusion equations and other interpretation methods based on wholly global quantities such as energy and entropy, this viewpoint helps us understand the filtering both globally (information reduction as image components shrink) and locally (how the image component contributes to its shrink rate). With careful consideration of the correspondence between eigenvalues and eigenvectors and their features, differences between linear and nonlinear filtering, as well as between isotropic and anisotropic filtering, are discussed. We also get some stability results of the filtering schemes. Several examples are provided to illustrate the properties.

Abstract. Fluid flow in underground formations is a nonlinear process. In this article we modelled the nonlinear transient flow behaviour of well production in an underground formation. Based on Darcy's law and material balance equations, we used quadratic pressure gradients to deduce diffusion equations and discuss the origins of nonlinear flow issues. By introducing an effective-well-radius approach that considers skin factor, we established a nonlinear flow model for both gas and liquid (oil or water). The liquid flow model was solved using a semi-analytical method, while the gas flow model was solved using numerical simulations because the diffusion equation of gas flow is a stealth function of pressure. For liquid flow, a series of standard log-log type curves of pressure transients were plotted and nonlinear transient flow characteristics were analyzed. Qualitative and quantitative analyses were used to compare the solutions of the linear and nonlinear models. The effect of nonlinearity upon pressure transients should not be ignored. For gas flow, pressure transients were simulated and compared with oil flow under the same formation and well conditions, resulting in the conclusion that, under the same volume rate production, oil wells demand larger pressure drops than gas wells. Comparisons between theoretical data and field data show that nonlinear models will describe fluid flow in underground formations realistically and accurately.

ABSTRACTThe effect of different analyte diffusion/adsorption protocols was studied which is often overlooked in surface-enhanced Raman scattering (SERS) technique. Three protocols: highly concentrated dilution (HCD) protocol, half–half dilution (HHD) protocol and layered adsorption (LA) protocol were studied and the SERS substrates were monolayer films of 80 nm Ag nanoparticles (NPs) which were modified by polyvinylpyrrolidone. The diffusion/adsorption mechanisms were modelled using the diffusion equation and the electromagnetic field distribution of two adjacent Ag NPs was simulated by the finite-different time-domain method. All experimental data and theoretical analysis suggest that different diffusion/adsorption behaviour of analytes will cause different SERS signal enhancements. HHD protocol could produce the most uniform and reproducible samples, and the corresponding signal intensity of the analyte is the strongest. This study will help to understand and promote the use of SERS technique in quantitative analysis.

Proton transfer in water–hydroxyl mixed overlayers on Pt(1 1 1): Combined approach of laser desorption and spatially-resolved X-ray photoelectron spectroscopy

We studied the applicability of classical scarp degradation modelling to active normal faults in the Lower Rhine Embayment. Our quantitative analysis was conducted on the frontal Bree fault scarp (Feldbiss fault) in Belgium and the Peel fault scarp near the city of Neer in the Netherlands. Vertical offset and diffusion age of these scarps have been modelled from elevation profiles across the studied faults using the diffusion equation. For that purpose, a computer-program (profil 2000) has been written, providing a sensitivity analysis of the determined parameters in function of the spatial repartition of the elevation measurements along the considered profiles. The results of this morphometric analysis have been validated by a comparison with the geologic record of the tectonic activity observed in the trenches excavated at the sites where the measurements have been conducted.We conclude that the modelling can only be applied to study tectonic activity since the Last Glacial Maximum (±14-19 kyr BP) because the surface expression of older paleoearthquakes in unconsolidated Late Pleistocene sediments has been erased by the strong erosive phase that occurred at the end of this glacial period. Even for Holocene scarps, morphologic dating seems very difficult because man-made perturbations destroyed surface evidence of the very recent fault activity in many sites. Nevertheless, we estimate that an appropriate value for the mass diffusivity constant for~ 1-m-high scarps in the investigated region is 0.002 to 0.010 m2/yr. On the other hand, vertical offsets can be determined with a good precision. These amount to respectively ~1 m and 1,3 m since the Last Glacial Maximum on the Feldbiss fault in Belgium and the Peel fault near Roermond in the Netherlands.

Summary The results of an attempt to utilize the agar diffusion technique of Oudin as a means of quantitatively evaluating the amount of antigen in a solution were presented. The linear relation between the logarithms of the antigen and antibody concentrations, and k, as reported by others, (3, 4) was confirmed, and empirical equations for egg albumin and bovine serum albumin systems were presented. The results of Becker et al. (4) were confirmed in that it was found possible to calculate the diffusion coefficient of these antigens by a derivative of Fick’s equation for diffusion, though only to a rough approximation. Further, the results obtained agreed excellently with those obtained for the same antigens by these authors. The value of the diffusion coefficient so calculated depended on the ratio of antigen to antibody, and the best results for the diffusion coefficient were found at the highest ratios. The effect of increasing saline and protein (non-cross-reacting) content on the migration of the specific precipitate ring was studied. It was found that increasing the saline or protein content of the medium in which the antigen was suspended increased the velocity of descent of the ring, and that the increase in velocity thus demonstrated reached a maximum at 2–2.5% saline, or protein, or total saline and protein, concentration.

Starting from the linear equations for reactive mechanical diffusion we propose an elementary mathematical model for high-temperature oxidation of metals in an isothermic regime. In the initial relations we take into account the fact that oxidation results in the formation of three characteristic zones: an oxide film, a transitional region, and undamaged material. We carry out a quantitative analysis and exhibit the laws of oxidation of steel pipes with mass-isolated inner surfaces.

A theoretical analysis has been developed to describe composition changes resulting from exposure of a unidirectional filament reinforced metal composite to an elevated temperature environment. The analysis was based on a simple superposition of finite-difference solutions of the diffusion equation, and was applicable to the case for which the filament and matrix metals comprised an isomorphous binary alloy system. The possibility of chemical reaction between filaments and matrix was excluded. Excellent agreement was obtained between calculated concentration profiles and results of quantitative electron microprobe analyses of nickel filament reinforced copper samples. However, the agreement was found to be quite sensitive to the dependence of diffusivity on concentration.

The dual effects of chemical composition and cooling rate on microstructural evolution were theoretically simulated extending the Johnson-Mehl equation for non-isothermal pro-eutectoid ferrite transformation kinetics incorporating the effect of supercooling (ΔT) as well as for isothermal pearlite transformation kinetics assuming the exponent on time ‘n’ to vary inversely with time as, n = p/tm, for case carburized steel quenched in oil. Carbon concentration profile of steel carburized at 930 °C for 10 h was theoretically computed by solving the Fick’s diffusion equation. The cooling curves from surface to core were generated for typical 20.32 and 15.6 mm diameter steels using FEM package (ANSYS) for oil quenching, water quenching, and air cooling. The effect of varying carbon concentration from surface to core was incorporated in the heat transfer equations while generating cooling curves, at different case depths (surface, 0.31, 0.558, 1.239, and 3.469 mm). The cooling curves for oil quenching were superimposed on the published TTT diagrams of the steels of corresponding carbon content and using the empirical equations the evolution of different microconstituents, e.g., ferrite, pearlite, and martensite from the parent austenite phase were computed for the carburized 20.32 and 15.6 mm diameter steel samples. These steel samples were also case carburized experimentally at 930 °C for 10 h followed by oil quenching. The theoretically predicted case depth of 3.469 mm matched closely with the experimentally observed value. Microstructural studies were done on inverted microscope and quantitative image analyzer at different case depths/nodal points. Microhardnesses were also measured at case depths from 0.1 mm to center of the samples at selected areas to identify the different phases. The experimentally observed microstructures matched well with the theoretically predicted evolution of microconstituents.

Fluorescent dyes are commonly used to study permeable (gap) junctions, but only rarely have quantitative values for junctional dye permeability been determined. In the present study, junctional permeance (PA, i.e., the product of the junctional permeability coefficient, P, times the junctional area, A) to Lucifer Yellow CH (LY) has been obtained for pairs of Novikoff hepatoma cells. Dye was microinjected into one cell and the subsequent transfer monitored by a SIT camera and recorded on video tape. The intensities of fluorescence in the injected and "recipient" cell were measured using a Digisector (Microworks) digitizing board and an Apple II Plus computer to analyze the video records. These changes in intensity, along with an estimate of volume of the spherical cells, were used to calculate the junctional permeance (PA) of cell pairs according to Fick's diffusion equation. Junctional permeances show considerable variation ranging from 0.08 X 10(-11) to 27.0 X 10(-11) cm3/sec. Using the mean PA and a previous estimate of the mean number of junctional channels per interface in the Novikoff cultures, a value for diffusion coefficient of LY through gap junctions is calculated to be about 1.4 X 10(-6) cm2/sec. There is a general proportionality between mean PA and cell volume for hepatoma cell pairs of a certain size range. Such a relationship between cell volume and junctional capacity suggests one source of variation of PA. Other possible sources, e.g., related to position in the cell cycle, are discussed.

Segmentation of regions of interest in an image has important applications in medical image analysis, particularly in computer aided diagnosis. Segmentation can enable further quantitative analysis of anatomical structures. We present efficient image segmentation schemes based on the solution of distinct partial differential equations (PDEs). For each known image region, a PDE is solved, the solution of which locally represents the weighted distance from a region known to have a certain segmentation label. To achieve this goal, we propose the use of two separate PDEs, the Eikonal equation and a diffusion equation. In each method, the segmentation labels are obtained by a competition criterion between the solutions to the PDEs corresponding to each region. We discuss how each method applies the concept of information propagation from the labelled image regions to the unknown image regions. Experimental results are presented on magnetic resonance, computed tomography, and ultrasound images and for both two-region and multi-region segmentation problems. These results demonstrate the high level of efficiency as well as the accuracy of the proposed methods.

Enormous percentage of total original oil in place lies in the fractured reservoirs. There dual permeability plays an important role to enhance the oil production rate which is governed by the large surface area provided by fractures so that fluids can move easily from low permeability zone matrix to high permeability channeled network zone. Convective current is setup in reservoir due to density inversion. This creates a favorable condition of mass transfer by diffusion. Due to high geo thermal gradient and convective motion of fluids in geothermal reservoir, diffusion can lead to the elevation or depression of bubble point pressure. When reservoir pressure gets drops it leads to evolution of gas from matrix zone which eventually gets transferred to gas cap. This leads to depression of Bubble Point. Now, when reservoir pressure increases it leads to transportation of gas from gas cap to matrix zone. This leads to elevation of Bubble point. In this paper a quantitative analysis of diffusion process for an arbitrary inclined fracture [1] by the help of FICK'S law of diffusion and continuity equation of diffusion is done. Through this analysis we obtain a final differential equation by applying material balance on the dissolved gas in the fracture. Lastly effective diffusion coefficient of the fractured media is quantitatively analyzed. Thus, correct analysis of reservoir in terms of microscopic and macroscopic sweep efficiency is done by the inclusion of this diffusion process. Hence, a better model to simulate the fractured reservoir can be made by selecting appropriate boundary conditions for the final differential equation obtained and proposed in this paper. Also we can understand the actual solution gas drive mechanism in the fractured reservoirs correctly. Negligible works have been done on the Mass Transfer analysis in the inclined fractured reservoir theoretically or experimentally and thus here an effective mathematical model of the mass transfer in an arbitrary inclined fracture has been formulated and presented for regulators of hydrocarbon industry for high outcome from a field.

The investigation of very slowly tumbling spin labels by nonlinear electron spin response techniques is discussed. Such techniques permit characterization of rotational processes with correlation times from 10(-3) to 10(-7) sec even though the linear spin response (ESR) technique is insensitive to motion in this region. Nonlinear techniques fall into two categories: (a) Techniques (referred to as passage techniques) in which the distribution of saturation throughout the spin system is determined both by the applied magnetic field modulation of the resonance condition and by the modulation of the resonance frequency induced by the molecular motion. The time dependence of this distribution produces phase and amplitude changes in the observed signals. (b) Techniques that measure the integral of the distribution function of the time required for saturated spin packets to move between pumped and observed portions of the spectrum [stationary and pulsed electron electron double resonance (ELDOR) techniques]. Quantitative analysis of passage ESR and stationary ELDOR techniques can be accomplished employing a density matrix treatment that explicitly includes the interaction of the spins with applied radiation and modulation fields. The effect of molecular motion inducing a random modulation of the anisotropic spin interactions can be calculated by describing the motion by the diffusion equation appropriate to the motional model assumed. For infinitesimal steps the eigen-functions of the diffusion operator are known analytically, while for random motion of arbitrary step size they are determined by diagonalizing the transition matrix appropriate for the step model used. The present communication reports investigation of the rotational diffusion of the spin label probes 2,2,6,6-tetramethyl-4-piperidinol-1-oxyl and 17beta-hydroxy-4',4'-dimethylspiro-[5alpha-androstane-3,2'-oxazolidin]-3'-oxyl in sec-butylbenzene. Experimental spectra are compared with computer simulations of spectra carried out for isotropic Brownian (limit of infinitesimal step size) and free diffusion (arbitrary step size) models.

The time scale of proton transfer between H(2)O and OH adsorbed on a Pt(111) surface was determined by a combination of laser-induced thermal desorption (LITD) and microscale x-ray photoelectron spectroscopy (micro-XPS). The patterned distribution OH+H(2)O/H(2)O/OH + H(2)O was initially prepared on the Pt(111) surface by the LITD method and the time evolution of the spatial distribution of H(2)O and OH was observed by the micro-XPS technique. From quantitative analyses based on a diffusion equation, we found two proton-transfer pathways with different time scales of 5.2+/-0.9 ns and 48+/-12 ns at 140 K, which were attributed to direct proton transfer to the neighbor site and H(3)O(+)-mediated transfer to the next-nearest site, respectively.

Crypt-OR: A privacy-preserving distributed cloud computing framework for object-removal in the encrypted images